Question

A bicyclist is at point A on a paved road and must ride to point C on another paved road. The two roads meet at

Answer 1

It would be faster for the bicyclist to ride from A to C on the paved roads since the time to go from A to C on the paved roads is 1.4 h and the time to go from A to C off-road is 1.7 h.        

To calculate which way would be faster we need to find the distance from point A to C with the law of cosines:

$\overline{AC}^{2} = \overline{AB}^{2} + \overline{BC}^{2} - 2\overline{AB}\overline{BC}cos(38)$

Where:

$\overline{AB}$: is the distance between the point A and B = 18 mi

$\overline{BC}$: is the distance between the point B and C = 12 mi        

$\overline{AC} = \sqrt{(18 mi)^{2} + (12 mi)^{2} - 2*18 mi*12 mi*cos(38)} = 11.3 mi$

Now, let's find the time for the two following cases.

1. From point A to C on the paved roads (t₁)

$t_{1} = t_{AB} + t_{BC}$

The time can be calculated with the following equation:

$t = \frac{d}{v}$    (1)

Where:

d: is the distance

v: is the velocity

Then, the total time that it takes the bicyclist to go from point A to C on the paved roads is:

$t_{1} = t_{AB} + t_{BC} = \frac{18 mi}{22 mph} + \frac{12 mi}{22 mph} = 1.4 h = 84 min$

2. From point A to C off-road (t₂)

With equation (1) we can calculate the time to go from point A to C off-road:

$t_{2} = \frac{\overline{AC}}{v_{2}} = \frac{11.3 mi}{6.8 mph} = 1.7 h = 102 min$

Therefore, it would be faster for the bicyclist to ride from A to C on the paved roads.  

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Answer 2

Answer:

Step-by-step explanation:

The diagrammatic expression to understand this question very well is attached in the image below.

By applying the law of cosine rule; we have:

• a² = b² + c² - 2bc Cos A --- (1)
• b² = a² + c² - 2ac Cos B --- (2)
• c² = a² + b² - 2ab Cos C --- (3)

From the diagram attached below, we need to determine the side "b" by using equation (2) from above:

b² = a² + c² - 2ac Cos B

From the information given:

a = 12 miles;  c = 18 miles;   ∠B = 38°

∴

replacing the values into the above equation:

b² = 12² + 18² - 2(12)(18) Cos (38°)

b² = 144 + 324 - 432 × (0.7880)

b² = 468 - 340.416

b² = 127.584

$b = \sqrt{127.584}$

b = 11.30 miles

However, we are also being told that the speed from A → C = 6.8 mph

Thus, the time required to go from A → C  can be determined by using the relation:

$\mathbf{speed = \dfrac{distance}{time}}$

making time the subject of the formula, we have:

$\mathbf{time= \dfrac{distance}{speed }}$

$\mathbf{time= \dfrac{11.30}{6.8}}$

time = 1.66 hours

By using the paved roads, the speed is given as = 22 mph

thus, the total distance covered = |AB| + |BC|

= (18+12) miles

= 30 miles

∴

$\mathbf{time= \dfrac{distance}{speed }}$

$\mathbf{time= \dfrac{30}{22}}$

time = 1.36 hours

Therefore, the time used off-road = 1.661 hours while the time used on the paved road is 1.36 hours.

Since we are considering the shortest time possible;

We can conclude that it would be faster for the bicyclist to ride from A to C on the paved roads since it takes a shorter time to reach its destination compared to the time used off-road.

Learn more about Law of cosine here:

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